Please help with this one. I've worked it out in so many ways and can't tell what's correct.
sinA sinA
----- - ------
1+sinA 1-sinA
Which answer is correct for the problem above?
a) 1+cotA
b) sinA tanA
c) -2tan^2A
d) secA cscA
common denominator: 1-sin^2 A
and that is equal to cos^2 A.
Numerator:
sinA-sin^2A - sinA -sin^2 A
-2sin^2 A
The result:
-2sin^2A /cos^2 A and you recall that tan u= sinU/cosU
To simplify the expression, we can find a common denominator for the two terms:
sinA / (1 + sinA) - sinA / (1 - sinA)
The common denominator for the two fractions is (1 + sinA)(1 - sinA), which simplifies to 1 - sin^2 A (using the difference of squares).
Now, let's simplify the numerators:
Numerator 1: sinA - sin^2 A
Numerator 2: -sinA - sin^2 A
Combining the numerators, we have: sinA - sin^2 A - sinA - sin^2 A
Simplifying, we get: -2sin^2 A
Now, let's rewrite the expression with the simplified numerator:
-2sin^2 A / (1 - sin^2 A)
Since 1 - sin^2 A is equal to cos^2 A (using the Pythagorean identity), we can substitute cos^2 A into the denominator:
-2sin^2 A / cos^2 A
Using the definition of tangent (tan A = sin A / cos A), we can rewrite this as:
-2(tan^2 A)
Therefore, the correct answer is c) -2tan^2 A.